867 research outputs found
Anchored burning bijections on finite and infinite graphs
Let be an infinite graph such that each tree in the wired uniform
spanning forest on has one end almost surely. On such graphs , we give a
family of continuous, measure preserving, almost one-to-one mappings from the
wired spanning forest on to recurrent sandpiles on , that we call
anchored burning bijections. In the special case of , ,
we show how the anchored bijection, combined with Wilson's stacks of arrows
construction, as well as other known results on spanning trees, yields a power
law upper bound on the rate of convergence to the sandpile measure along any
exhaustion of . We discuss some open problems related to these
findings.Comment: 26 pages; 1 EPS figure. Minor alterations made after comments from
refere
A Guide to Stochastic Loewner Evolution and its Applications
This article is meant to serve as a guide to recent developments in the study
of the scaling limit of critical models. These new developments were made
possible through the definition of the Stochastic Loewner Evolution (SLE) by
Oded Schramm. This article opens with a discussion of Loewner's method,
explaining how this method can be used to describe families of random curves.
Then we define SLE and discuss some of its properties. We also explain how the
connection can be made between SLE and the discrete models whose scaling limits
it describes, or is believed to describe. Finally, we have included a
discussion of results that were obtained from SLE computations. Some explicit
proofs are presented as typical examples of such computations. To understand
SLE sufficient knowledge of conformal mapping theory and stochastic calculus is
required. This material is covered in the appendices.Comment: 80 pages, 22 figures, LaTeX; this version has 5 minor corrections to
the text and improved hyperref suppor
Algorithms for sampling spanning trees uniformly at random
Thesis on the analysis of various algorithms for sampling spanning trees of a graph uniformly at random
SLE for theoretical physicists
This article provides an introduction to Schramm(stochastic)-Loewner
evolution (SLE) and to its connection with conformal field theory, from the
point of view of its application to two-dimensional critical behaviour. The
emphasis is on the conceptual ideas rather than rigorous proofs.Comment: 43 pages, to appear in Annals of Physics; v.2: published version with
minor correction
Field theoretic formulation and empirical tracking of spatial processes
Spatial processes are attacked on two fronts. On the one hand, tools from theoretical and
statistical physics can be used to understand behaviour in complex, spatially-extended
multi-body systems. On the other hand, computer vision and statistical analysis can be
used to study 4D microscopy data to observe and understand real spatial processes in
vivo.
On the rst of these fronts, analytical models are developed for abstract processes, which
can be simulated on graphs and lattices before considering real-world applications in elds
such as biology, epidemiology or ecology. In the eld theoretic formulation of spatial processes,
techniques originating in quantum eld theory such as canonical quantisation and
the renormalization group are applied to reaction-di usion processes by analogy. These
techniques are combined in the study of critical phenomena or critical dynamics. At this
level, one is often interested in the scaling behaviour; how the correlation functions scale
for di erent dimensions in geometric space. This can lead to a better understanding of how
macroscopic patterns relate to microscopic interactions. In this vein, the trace of a branching
random walk on various graphs is studied. In the thesis, a distinctly abstract approach
is emphasised in order to support an algorithmic approach to parts of the formalism.
A model of self-organised criticality, the Abelian sandpile model, is also considered. By
exploiting a bijection between recurrent con gurations and spanning trees, an e cient
Monte Carlo algorithm is developed to simulate sandpile processes on large lattices.
On the second front, two case studies are considered; migratory patterns of leukaemia cells
and mitotic events in Arabidopsis roots. In the rst case, tools from statistical physics
are used to study the spatial dynamics of di erent leukaemia cell lineages before and after
a treatment. One key result is that we can discriminate between migratory patterns in
response to treatment, classifying cell motility in terms of sup/super/di usive regimes.
For the second case study, a novel algorithm is developed to processes a 4D light-sheet
microscopy dataset. The combination of transient uorescent markers and a poorly localised
specimen in the eld of view leads to a challenging tracking problem. A fuzzy
registration-tracking algorithm is developed to track mitotic events so as to understand
their spatiotemporal dynamics under normal conditions and after tissue damage.Open Acces
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